Finite Group Databasebeta

← #10695892

Subgroups — #10695892

Order 768.

562 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78641C211yesmaximal, characteristic
2C2×#4735C211yesmaximal, characteristic, complemented
3C2×#4736C211yesmaximal, characteristic, complemented
4#69973C211yesmaximal, characteristic, complemented
5#69921C211yesmaximal, characteristic, complemented
6#69921C211yesmaximal, characteristic, complemented
7C3×#1298C211yesmaximal, characteristic, complemented
8#9992—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C22×C4⋊Dic3C2211yescharacteristic
12C3×#407C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14C2×C3×C22⋊C8C2211yescharacteristic
15#5207C2211yescharacteristic
16#5135—22no—
17C2×#821—21no—
18C2×#821—21no—
19C2×#818—21no—
20C2×#817—21no—
21#4736—22no—
22#4736—22no—
23#4735—21no—
24#4735—21no—
25#4735—22no—
26#4735—22no—
27#4732—21no—
28#4732—21no—
29#4554—22no—
30#4468—22no—
31#4468—22no—
32#3192—31no—
33C2×#481—31no—
34C2×#480—31no—
35#1821—31no—
36#1471—31no—
37#1471—31no—
38#1298D311yescharacteristic, complemented
39C23×C3×C4C2311yescharacteristic
40C22×#67—21no—
41C22×#67—21no—
42C23×Dic3—21no—
43C22×D12D411yescharacteristic
44C22×D12D411yescharacteristic
45C2×C3×M16D412yes—
46C22×C3×C8—21no—
47C22×C3×C8—21no—
48#930—21no—
49#930—21no—
50#930—21no—
51#930—21no—
52#930—21no—
53#930—21no—
54#930—21no—
55#930—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3D412yes—
64C2×C4⋊Dic3D411yescharacteristic
65C2×C4⋊Dic3D411yescharacteristic
66C2×#208—21no—
67C2×D24—22no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8D412yes—
73C3×C22⋊C8D412yes—
74#830—21no—
75#830—21no—
76#821—22no—
77#821—22no—
78#821—24no—
79#821—24no—
80#818—24no—
81#817—22no—
82#817—24no—
83C2×#121—31no—
84C2×#121—31no—
85C22×C4⋊C4—31no—
86#481—62no—
87#481—61no—
88#481—61no—
89#481—62no—
90#480—62no—
91#480—62no—
92#469—62no—
93#447—61no—
94#447—61no—
95C2×#107—61no—
96C2×#106—61no—
97C2×#102—61no—
98C2×#102—61no—
99#407—31no—
100#407C2×D311yescharacteristic
101C2×C22⋊C8C2×D311yescharacteristic
102C2×C22⋊C8C2×D311yescharacteristic
103#343—62no—
104#323—62no—
105#323—62no—
106C24×C3#4111yescharacteristic
107C23×D3—21no—
108C23×D3—21no—
109C22×C3×C4—42no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C2×#67—41no—
117C2×#67—41no—
118C2×#67—41no—
119C2×#67—41no—
120C2×#67—82no—
121C2×#67—82no—
122C22×Dic3—22no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125C22×Dic3—41no—
126C22×Dic3—41no—
127C22×Dic3—82no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—42no—
133C2×D12—42no—
134C3×M16—42no—
135C2×C3×C8—22no—
136C2×C3×C8—22no—
137C2×C3×C8—22no—
138C2×C3×C8—42no—
139C2×C3×C8—42no—
140(C2×D3)⋊C4—41no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—41no—
143(C2×D3)⋊C4—41no—
144C4⋊Dic3—21no—
145C4⋊Dic3—22no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—22no—
150C4⋊Dic3—41no—
151C4⋊Dic3—41no—
152#208—42no—
153C2×C3⋊C8—41no—
154D24—82no—
155C22×D4—31no—
156C22×D4—31no—
157C22×D4—61no—
158C22×D4—61no—
159C23×C4—61no—
160C23×C4C22×D311yescharacteristic
161C2×D8—62no—
162C2×M16—61no—
163C2×M16D1212yes—
164C22×C8—21no—
165C22×C8—21no—
166#121—61no—
167#121—61no—
168#121—61no—
169#121—61no—
170#121—61no—
171#121—61no—
172#121—61no—
173#121—61no—
174C2×C4⋊C4—32no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—61no—
184#107—64no—
185#106—64no—
186#106—62no—
187#102—64no—
188#102—64no—
189#102—62no—
190#102—62no—
191C22⋊C8—21no—
192C22⋊C8—21no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195C22⋊C8—61no—
196C22⋊C8—61no—
197C22⋊C8D1212yes—
198C22⋊C8D1212yes—
199C23×C3—22no—
200C23×C3—22no—
201C23×C3—41no—
202C23×C3#12111yescharacteristic
203C23×C3#12111yescharacteristic
204C23×C3#12111yescharacteristic
205C22×D3—41no—
206C22×D3—41no—
207C22×D3—42no—
208C22×D3—42no—
209C22×D3—81no—
210C22×D3—81no—
211C22×D3—81no—
212C22×D3—81no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12711yescharacteristic
223#67—82no—
224#67—82no—
225#67—82no—
226#67—82no—
227C2×Dic3—42no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—82no—
231C2×Dic3—82no—
232C2×Dic3—82no—
233C2×Dic3—81no—
234C2×Dic3—81no—
235D12—42no—
236D12—42no—
237D12—42no—
238D12—42no—
239C3×C8—42no—
240C3×C8—42no—
241C3×C8—42no—
242C3⋊C8—81no—
243C24—61no—
244C24—61no—
245C24#23911yescharacteristic
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—122no—
251C2×D4—122no—
252C2×D4—121no—
253C2×D4—121no—
254C2×D4—121no—
255C2×D4—121no—
256C2×D4—242no—
257C2×D4—242no—
258C22×C4—42no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4—62no—
262C22×C4—121no—
263C22×C4—121no—
264C22×C4—242no—
265C22×C4C2×D1211yescharacteristic
266C22×C4C2×D1211yescharacteristic
267C22×C4C2×D1211yescharacteristic
268C22×C4D3×D411yescharacteristic
269C22×C4D3×D411yescharacteristic
270C22×C4D3×D411yescharacteristic
271D8—242no—
272M16—42no—
273M16—122no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—121no—
280C4⋊C4—62no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—62no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—121no—
287C4⋊C4—121no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C22⋊C4—121no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—44no—
300C22×C3#39211yescharacteristic
301C22×C3#47112yes—
302C22×C3#47212yes—
303C2×D3—81no—
304C2×D3—81no—
305C2×D3—81no—
306C2×D3—81no—
307C2×D3—81no—
308C2×D3—81no—
309C2×D3—162no—
310C2×D3—162no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4#45812yes—
315Dic3—82no—
316Dic3—81no—
317Dic3—81no—
318C23—22no—
319C23—22no—
320C23—41no—
321C23—121no—
322C23—121no—
323C23—122no—
324C23—122no—
325C23—241no—
326C23—241no—
327C23—241no—
328C23—241no—
329C23#89511yescharacteristic
330C23#89511yescharacteristic
331C23#89511yescharacteristic
332D4—122no—
333D4—122no—
334D4—122no—
335D4—122no—
336D4—242no—
337D4—242no—
338D4—242no—
339D4—242no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—122no—
349C2×C4—242no—
350C2×C4—242no—
351C2×C4—242no—
352C2×C4—241no—
353C2×C4—241no—
354C2×C4#87411yescharacteristic
355C2×C4#88211yescharacteristic
356C2×C4#88211yescharacteristic
357C2×C4#88211yescharacteristic
358C8—42no—
359C8—42no—
360C8—42no—
361C8—241no—
362C2×C3—22no—
363C2×C3—22no—
364C2×C3—41no—
365C2×C3#201811yescharacteristic
366C2×C3#201911yescharacteristic
367C2×C3#202211yescharacteristic
368D3—161no—
369D3—161no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—44no—
378C22—241no—
379C22—241no—
380C22—241no—
381C22—241no—
382C22—241no—
383C22—241no—
384C22—482no—
385C22—482no—
386C22#467111yescharacteristic
387C22#489712yes—
388C22#489812yes—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—242no—
393C4—241no—
394C4—241no—
395C4#474212yes—
396C3#999211yescharacteristic, complemented, Sylow
397C2—22no—
398C2—22no—
399C2—41no—
400C2—481no—
401C2—481no—
402C2#6894211yescharacteristic
403C2#6894311yescharacteristic
404C2#6896611yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1298C2
socleC22×C3#392
prime radical#78641C2
hypercenter#1298D3
hypocenterC3#9992
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1298

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#4735C2semidirect
3. C2×#4736C2semidirect
4. #69973C2semidirect
5. #69921C2semidirect
6. #69921C2semidirect
7. C3×#1298C2semidirect
38. #1298D3semidirect
396. C3#9992semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#786412
C2×#47352
C2×#47362
#699732
#699212
C3×#12982
#99923